Acta mathematica scientia,Series A ›› 2019, Vol. 39 ›› Issue (4): 909-917.

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Global Stability of a Measles Epidemic Model with Partial Immunity and Environmental Transmission

Xiaojie Jing,Aimin Zhao*(),Guirong Liu   

  1. School of Mathematical Sciences, Shanxi University, Taiyuan 030006
  • Received:2018-04-12 Online:2019-08-26 Published:2019-09-11
  • Contact: Aimin Zhao
  • Supported by:
    国家自然科学基金;山西省自然科学基金;the NSFC;the Natural Science Foundation Program of ShanxiProvince


In this paper, a measles epidemic model with partial immunity and environmental transmission is considered, and the basic reproduction number R0 is obtained. By constructing Lyapunov functions, we prove the global asymptotic stability of the infection-free equilibrium and the endemic equilibrium. When R0 < 1, the infection-free equilibrium is globally asymptotically stable, which implies that measles dies out eventually; when R0 > 1, the model has a unique endemic equilibrium, which is globally asymptotically stable, that is the transmission of measles keeps a steady state. Finally, the simulations are carried to verify the rationality of the results. This work has practical significance for guiding us to prevent and control the measles spread.

Key words: Partial immunity, Environmental transmission, The basic reproduction number, Lyapunov function, Global stability

CLC Number: 

  • O175.1